10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS 391
change. Also, for each function u e *C(K), we define Tn(u) on EM (15.10.1), and
by virtue of (15.10.2) we have \\Tn(u)\\ g \\u\\ for each n.
Now we have the following lemma:
(15.10.8.1) Let E be a Hilbert space which is the Hilbert sum of a sequence
(En) of closed subspaces. For each n let Un be a continuous operator on Ert, and
suppose that the sequence of norms (\\Un\\) is bounded. Then there exists a
unique continuous operator U on "E whose restriction to En is Un for each n.
Also, the restriction of U* to En is U*.
Suppose that || Un\\ ^a for all n. For each x = £ xn e E, where xn e En for
all n and ||x||2 = £ \\xn\\2 (6.4), we have
which shows that the series ]£ Un * xn converges in E. If U • x denotes its sum,
n
it is clear that U is linear and that, from above, || U-x\\ g a\\x\\, so that U is a
continuous operator (5.5.1). The uniqueness of U follows from the fact that
the union of the subspaces En is a total set in E (6.4). Finally, we have
|| jy*|| = || un\\ g a for all n, and therefore there exists a continuous operator V
on E whose restriction to En is U* for all n.lfy = ^yn, with yn e En for all n
and X \\yn\\2 = \\y\\2, then (6.4) we have
which proves that F= C7*.
Applying (15.10.8.1), we see that there exists a unique normal continuous
operator T(u) on E whose restriction to En is Tn(w), for each n. It is immediate
that wi— > T(u) is a representation of ^C(K) in E which extends the representa-
tion T of #C(K). Next, the proposition (15.10.4) generalizes without any
change in the proof: we have only to observe that we can take as a total subset
of E the set of all Tn(s) • an , where s e #C(K) anc* n ^s arbitrary.
In general there exist infinitely many decompositions of E as a Hilbert sum
of subspaces with the properties of (15.10.8). However, there is the following
result:
(15.10.9) There exists a decomposition ofE as a Hilbert sum of a (finite or
infinite) sequence (En) of closed subspaces, stable with respect to T, such that theuch that the restriction Tn of T to EB admits a totalizing vector an , for